Root polytopes and abelian ideals

TL;DR

This paper explores the structure of root polytopes associated with crystallographic root systems, linking their faces to abelian ideals and providing triangulations for types A and C.

Contribution

It establishes a connection between root polytopes and abelian ideals, and constructs explicit triangulations for specific root system types.

Findings

Determined hyperplane arrangements for faces of codimension 2.

Linked face structure with abelian ideals and Weyl group actions.

Provided triangulations for types A and C root polytopes.

Abstract

We study the root polytope $\mathcal P_\Phi$ of a finite irreducible crystallographic root system $\Phi$ using its relation with the abelian ideals of a Borel subalgebra of a simple Lie algebra with root system $\Phi$. We determine the hyperplane arrangement corresponding to the faces of codimension 2 of $\mathcal P_\Phi$ and analyze its relation with the facets of $\mathcal P_\Phi$. For $\Phi$ of type $A_n$ or $C_n$, we show that the orbits of some special subsets of abelian ideals under the action of the Weyl group parametrize a triangulation of $\mathcal P_\Phi$. We show that this triangulation restricts to a triangulation of the positive root polytope $\mathcal P_\Phi^+$.